Equivalent block diagrams, how to infer?

Question

Assume I have one block diagram and I want to simplify it to an equivalent block diagram, how do I find such block diagram apriori?

For example, Smith Predictor structure

The first pic above is Smith Predictor structure, and I would like to know how apriori I can infer that the pic below it is equivalent to it?

If I calculate the ratio of both I get C(s)/R(s) = G(s)F(s)H(s)/(1+G(s)F(s)), but that doesn't show that they are equivalent, I mean how do I know how the equivalent diagram looks like by only knowing this ratio?

Thanks in advance.

Answer 1

Here's a tip, simplify by visual inspection. Look at the two final blocks of H(s) and F(s). Can you see that if you switched the order that those two occured, you don't need the F(s) block in the short feedback loop.

Then if you looked at the polarity of H(s) being fed back you can see that all those terms cancel out - just look at the signs on the adders: -

X (input to G(s)) = -H(s) - 1 + H(s) = -1 i.e. there is nothing about H(s) that gets fed back and iterated: -